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Cones and ping-pong in three dimensions

Gabriel Frieden, Félix Gélinas and Étienne Soucy

Vol. 17 (2024), No. 1, 11–28
Abstract

We study the hypergeometric group in GL 3() with parameters α = (1 4, 1 2, 3 4) and β = (0,0,0). We give a new proof that this group is isomorphic to the free product 4 2 by exhibiting a ping-pong table. Our table is determined by a simplicial cone in 3, and we prove that this is the unique simplicial cone (up to sign) for which our construction produces a valid ping-pong table.

Keywords
hypergeometric group, free product of groups, ping-pong lemma
Mathematical Subject Classification
Primary: 20E06
Milestones
Received: 25 January 2022
Revised: 22 November 2022
Accepted: 17 December 2022
Published: 15 March 2024

Communicated by Jim Haglund
Authors
Gabriel Frieden
Université du Québec à Montréal
Montréal, QC
Canada
Félix Gélinas
Université du Québec à Montréal
Montréal, QC
Canada
Étienne Soucy
Université du Québec à Montréal
Montréal, QC
Canada